Аңдатпа
In this note we obtain Hardy and critical Hardy inequalities with any homogeneous quasi-norm in unified way. Actually, we show a sharp remainder formula for these results. In particular, our identity implies corresponding Hardy and critical Hardy inequalities with any homogeneous quasi-norm for the radial derivative operator, thus yielding improved versions of corresponding classical counterparts. Moreover, we discuss extensions of these results in the setting of Folland and Stein’s homogeneous Lie groups. Such a more general setting is convenient for the distillation of those results of harmonic analysis depending only on the group and dilation structures, which is one of our motivations working in the setting. Our approach based on the factorization method of differential operators introduced by Gesztesy and Littlejohn. As an application, we show Caffarelli-Kohn-Nirenberg type inequalities with more general weight. Because of the freedom in the choice of any homogeneous quasi-norm, our results give new insights already in both anisotropic ℝn and isotropic ℝn .